[Paper Review] Lieb-Schultz-Mattis, Luttinger, and 't Hooft -- anomaly matching in lattice systems

Meng Cheng, Nathan Seiberg|arXiv (Cornell University)|Nov 22, 2022

Cold Atom Physics and Bose-Einstein Condensates22 citations

TL;DR

The paper develops a unified anomaly-based framework for 1+1d lattice systems by coupling to background gauge fields and twisting boundary conditions, revealing how ’t Hooft anomalies constrain lattice symmetries, LSM-type results, and Luttinger constraints.

ABSTRACT

We analyze lattice Hamiltonian systems whose global symmetries have 't Hooft anomalies. As is common in the study of anomalies, they are probed by coupling the system to classical background gauge fields. For flat fields (vanishing field strength), the nonzero spatial components of the gauge fields can be thought of as twisted boundary conditions, or equivalently, as topological defects. The symmetries of the twisted Hilbert space and their representations capture the anomalies. We demonstrate this approach with a number of examples. In some of them, the anomalous symmetries are internal symmetries of the lattice system, but they do not act on-site. (We clarify the notion of "on-site action.") In other cases, the anomalous symmetries involve lattice translations. Using this approach we frame many known and new results in a unified fashion. In this work, we limit ourselves to 1+1d systems with a spatial lattice. In particular, we present a lattice system that flows to the $c=1$ compact boson system with any radius (no BKT transition) with the full internal symmetry of the continuum theory, with its anomalies and its T-duality. As another application, we analyze various spin chain models and phrase their Lieb-Shultz-Mattis theorem as an 't Hooft anomaly matching condition. We also show in what sense filling constraints like Luttinger theorem can and cannot be viewed as reflecting an anomaly. As a by-product, our understanding allows us to use information from the continuum theory to derive some exact results in lattice model of interest, such as the lattice momenta of the low-energy states.

Motivation & Objective

• Clarify how ’t Hooft anomalies manifest in lattice systems with discrete spatial structure.
• Show how twists/defects encode background gauge fields and reveal symmetry mixing and projective representations.
• Demonstrate anomaly matching in lattice models corresponding to known continuum results (e.g., c=1 compact boson).
• Reframe LSM-type theorems and Luttinger constraints within the anomaly-matching paradigm.

Proposed method

• Couple lattice systems to flat background gauge fields for internal symmetries via twisted boundary conditions.
• Insert spatial twists and translate them into topological defects to study symmetry actions on twisted Hilbert spaces.
• Compute partition functions with symmetry insertions to detect phase ambiguities under gauge transformations (anomalies).
• Interpret anomalies through anomaly inflow by extending to a higher-dimensional bulk as a mathematical device.
• Use continuum limits (like the c=1 compact boson) to guide lattice constructions that preserve full continuum-like symmetries and anomalies.
• Analyze lattice translations as emanant/internal symmetries and relate LSM and Luttinger-type constraints to anomaly considerations.

Experimental results

Research questions

• RQ1 How do ’t Hooft anomalies of internal symmetries manifest when lattice translations are present and the microscopic action is not on-site?
• RQ2 How can twisted boundary conditions and topological defects capture background gauge fields for lattice systems with anomalies?
• RQ3 Can LSM-type constraints and Luttinger’s theorem be understood as anomaly matching or related emanant symmetries in lattice models?
• RQ4 What lattice constructions flow to known continuum theories (e.g., c=1 compact boson) while preserving anomalies and dualities?
• RQ5 How do emergent/emanant symmetries arising from lattice translations participate in anomaly structures and spectral constraints?

Key findings

• Presented a lattice system that flows to the c=1 compact boson for every radius, preserving the full continuum internal symmetry and its anomalies, including T-duality.
• Demonstrated examples where anomalous internal symmetries do not act on-site on the lattice but can be analyzed via flat background gauge fields and twisted boundary conditions.
• Recast the LSM theorem as an ’t Hooft anomaly matching condition in spin chains, and extracted exact finite-size spectrum information using continuum insights.
• Identified emanant (translation-induced) Z2 symmetries in certain XXZ-type chains and explained their exactness at low energies despite lattice origins.
• Clarified when Luttinger-type filling constraints reflect anomalies and when they do not, highlighting a nuanced relationship between filling, emanant symmetries, and anomalies.